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Remark 2.5 �� Global minimizers of the energy in the continuum setting might be a convex combination of a finite number of Dirac deltas. Numerical experiments suggest that this is always the case in the range 2��>0, using uniform bounds on the ��th moments of the minimizers. First, if is selleck chemicals llc a minimizer of En with centre of mass at the origin, then by (2.3) and H?lder inequality or equivalently Because the function ��(x)=�Ox?y�O�� is convex for any �á�1 and , Jensen's inequality implies that As a consequence, we obtain a uniform bound on the ��th moment of the discrete minimizers. As a result, the minimizing sequence corresponding to the associated atomic measures is tight, leading also towards a global minimizer of E[��]. 3.?Further remarks in one dimension Here, we concentrate on the one-dimensional case (d=1) for more refined properties. (a) Confinement of discrete global minimizers We check first how sharp are the conditions on the exponents of the potential to obtain the confinement of global discrete minimizers independently of n. In fact, when the potential is very repulsive at the origin, we can show that a uniform bound in n of the diameter of the global minimizers Etomidate in the discrete setting does not hold. If x is a minimizer of En(x), we always assume that the labelling of the xi is C646 datasheet in increasing order: x1��x2�ܡ�xn. Theorem 3.1 �� Suppose ��0. Denote by an the unique element of such that 3.1 If x is a minimizer of En, we claim that 3.2 Indeed, otherwise, and we know that in this case En